Problem: Solve for $x$ : $ 6|x - 8| + 1 = 4|x - 8| + 3 $
Solution: Subtract $ {4|x - 8|} $ from both sides: $ \begin{eqnarray} 6|x - 8| + 1 &=& 4|x - 8| + 3 \\ \\ { - 4|x - 8|} && { - 4|x - 8|} \\ \\ 2|x - 8| + 1 &=& 3 \end{eqnarray} $ Subtract ${1}$ from both sides: $ \begin{eqnarray} 2|x - 8| + 1 &=& 3 \\ \\ { - 1} &=& { - 1} \\ \\ 2|x - 8| &=& 2 \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{2|x - 8|} {{2}} = \dfrac{2} {{2}} $ Simplify: $ |x - 8| = 1$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 8 = -1 $ or $ x - 8 = 1 $ Solve for the solution where $x - 8$ is negative: $ x - 8 = -1 $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& -1 \\ \\ {+ 8} && {+ 8} \\ \\ x &=& -1 + 8 \end{eqnarray} $ $ x = 7 $ Then calculate the solution where $x - 8$ is positive: $ x - 8 = 1 $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& 1 \\ \\ {+ 8} && {+ 8} \\ \\ x &=& 1 + 8 \end{eqnarray} $ $ x = 9 $ Thus, the correct answer is $x = 7 $ or $x = 9 $.